The tenor saxophonist Tommy Whittle, who has died aged 87, was as relaxed in the demanding session world as he was on the jazz bandstand. Having earned his spurs in dance-hall bands as a youngster, he became a member of the Ted Heath orchestra in the late 1940s. In parallel, he kept the jazz flame alive, always finding time to play club dates and running a series of successful jazz clubs involving the best local musicians. A "polished stylist", in the words of the cornettist Digby Fairweather, Whittle at his best was arguably one of Britain's finest jazz musicians.

Once his commercial assignments had diminished, Whittle returned to full-time jazz performance with his second wife, the singer Barbara Jay, whom he married in 1967. His desire to play and willingness to travel continued until failing health intervened late last year.

Whittle was born in Grangemouth, Scotland, and, after some family prompting, tried the harmonica and the banjo before alighting on the clarinet. Introduced by the painter Alan Davie, then a student, to the local rhythm club, he began to attend jam sessions and listen to recordings by Lester Young and Coleman Hawkins.

When Davie went into the army, Whittle borrowed his saxophone. Evidently a quick learner, he played briefly with a local band before his parents sent him to stay with his grandparents in Chatham, Kent, hoping to wean their 16-year-old son away from professional music by finding him a serious job. But a chance encounter with the drummer Ronnie Verrell led to a booking with the Claude Giddins band in Gillingham, which often featured guest stars from London, including the expatriate Belgian trumpeter Johnny Claes.

After playing with Claes for a few weeks, Whittle joined the veteran bandleader Lew Stone. On the London freelance scene, he performed with the reedman Carl Barriteau and then with Harry Hayes at Churchill's Club, a combo that included the pianist George Shearing.

Already making a name for himself as a soloist in classy circles, Whittle joined the Heath band in 1947, replacing Ronnie Scott, and stayed with this high-profile, well-paid outfit for the next five years. They did road trips, frequent recording dates, London Palladium concerts and an appearance in the 1950 movie Dance Hall. It was when Heath insisted that he repeat the same solos every night that Whittle made his bid for freedom by playing in the drummer Tony Kinsey's trio at Studio 51 in Soho.

He was seduced away from full-time jazz when Cyril Stapleton approached him in 1952 to join the newly formed BBC Show Band, thus putting him in a position to win both the Melody Maker and New Musical Express jazz polls. "I ended up doing three broadcasts a week and still maintaining my jazz club work," he said. After three years, keen to follow up on this success, Whittle formed his own 10-piece band, a starry ensemble that included the trumpeter Kenny Wheeler and the baritone saxist Joe Temperley, touring continuously for the next year or so. Various Whittle groups, quartets or sextets also toured the US, including one package with Chuck Berry and the Platters. Whittle guested with the Stan Kenton band on their 1956 UK tour, replacing an errant sideman who had been dispatched back to the US.

Thereafter, Whittle was immersed in commercial music – first leading his own group at the Dorchester hotel, with the tenor saxophonist Bobby Wellins alongside – interspersed with session calls and followed by a substantial period with the Jack Parnell TV orchestra, based at Elstree Studios.

Committed to constant practice, Whittle became proficient on an arsenal of woodwind instruments, ready for whatever session calls came his way. Yet he also managed to keep a highly creditable jazz career going, operating his own club at the Hop Bine pub in Wembley, often jousting with visiting star soloists, including fellow saxophonists Tubby Hayes and Tony Coe.

Eventually, the Hop Bine changed hands and Barbara devised the Ella Fitzgerald Songbook show for a trio of singers including Barbara, supported by Tommy's quartet. It was extremely successful. In parallel, the quietly spoken Whittle led the Pizza Express All-Stars, played guest spots with the Ray McVay orchestra and was always on hand for any Heath band revivals, while recording regularly with his own groups, his playing as distinctive and hard-swinging as it had always been.

He is survived by Barbara and his sons, Sean, a pianist and composer, and Martin, from his first marriage.

• Thomas Whittle, jazz saxophonist, born 13 October 1926; died 13 October 2013